ON THE FOURIER DEVELOPMENTS OF A CERTAIN
CLASS OF THETA QUOTIENTS
M. A. BASOCO
1. Introduction. In this paper we shall be concerned with the functions <t>a(z) defined by the relation
(1)
k
(à
*«(*) s 2—log#a(zt
\
q)\
k
(û ' (z a) \ k
= <* / \ > ,
a = 0, 1, 2, 3,
where ûa(z, q) is a Jacobi theta function and k is a positive integer.
In the first place, we shall derive the Fourier developments which
represent these functions in a certain strip of the complex plane;
it will be seen that the Fourier coefficients of <t>*a(z) depend on those
of 4>*a(z)i $ = 1 , 2 , 3 , • • • , & — 1, through a recurrence relation of order
k. Secondly, these developments, in conjunction with certain obvious
identities, yield, through the method of paraphrase, some general
arithmetical formulae of a type first given by Liouville. 1 Indeed, we
recover, in a simple manner, some results given without proof by
Liouville, which were later proved by Bell2 through the use of somewhat complex identities involving a certain set of doubly periodic
functions of the second kind. One of these results has recently been
proved in a strictly elementary, but very ingenious way, by Uspensky. 8 Finally, we indicate some applications of these formulae to the
derivation of a certain type of arithmetic and algebraic identities.
2. The functions <f>%(z). It should be pointed out that the case k = l9
is implicit in §§47 and 48 of Jacobi's Fundamenta nova.4 Likewise, the
case k = 2, has been obtained by G. D. Nichols 5 through the use of
certain results due to the present writer. 6 The following is a direct
derivation of the necessary procedure for the general case; it depends
on a straightforward application of contour integration and the theory of residues. It is convenient to treat the two functions <f>l(z) and
Presented to the Society, November 28, 1942; received by the editors June 29,
1942.
1
J. Math. Pures Appl. (2) vol. 3 (1858) et seq. See, for example, vol. 3 p. 247 (H).
2
Trans. Amer. Math. Soc. vol. 22 (1921) p. 215 formula (xiv').
3
J. V. Uspensky and M. A. Heaslet, Elementary number theory, New York, 1939.
See chap. 13 p. 462 formula (R).
4
Jacobi, Gesammelte Werke, vol. 1 p. 187.
5
G. D. Nichols, Tôhoku Math. J. vol. 40 (1935) pp. 252-258.
« M. A. Basoco, Bull. Amer. Math. Soc. vol. 38 (1932) pp. 560-568.
299
300
M. A. BASOCO
[April
<f>*(z) separately; from the results for these functions we may obtain
the corresponding ones for a = 2, 3 by merely replacing z by z + ir/2.
It follows from the properties of the the ta functions 7 and Fourier's
theorem for analytic functions that
k
A
<t>o(z) = 2^ n
(2)
e
,
AT =
(1/T)
-
3(TT/2)
n=—oo
where,
f
(3)
T 2
/
I
To evaluate this integral, consider
the contour C being the boundary
at 2 = ±7r/2 and z — ±7T/2+TTT/2.
integral leads without difficulty to
7
«
k
I
•
2mz
<t>Q\z)e
dz.
the contour integral fc<l>o(z)e2nizdzt
of the parallelogram with vertices
Cauchy's theorem applied to this
the recurrence relation
je i
(4)
A
- q 2^CktA-
2t) An
= 2tR,
q == exp WIT,
3=0
where Ckj is the binomial coefficient k\/j\(k —j) ! and R is the residue
of the integrand at the pole of order k with affix z = TTT/2.
We thus obtain, on placing & = 1, 2, 3 and computing the necessary
residues, the following expressions for the coefficients in the expansions of <j*o(z)> for the values of k indicated:
(a)
Aü
m
(b) A
A(S)
(c)
Ao
- n.
J(1)
A{1)
8
_ j*».
^ -"
( 1
= U,* ^4n =
_
z^"
1 - q^
n ^ 0,
4(n - 1)
?3
? 2»)2
A-.n =
(2)
2n
1 - q
û{"
2iqn
&{
1 - q2n
2iqn
+ (18 - 4n2)22w + (2ri
6n + 3)}.
With respect to the function 4>\{z) the process used in the preceding
must be modified slightly on account of the presence of a singularity
(pole of order k) at the origin. Define the function xf/^ (z) to be of
the form
7
See Whittaker and Watson, Modern analysis, chap. 21.
i943]
CLASS OF THETA QUOTIENTS
301
(5)
*?\z) - <h{z) - T("\z),
where T(k)(z) is a suitable function of sin z and cos z, having the period
7T and such that the principal part of its Laurent expansion about the
origin coincides with that of <j>l(z). The function ^(z) is, therefore,
analytic along the real axis. I t is found that, for the cases considered
(that is, for & = 1, 2, 3), the functions T(k)(z) are of the form:
T™(z) = cot z',T^(z)
= cot2 z;T^\z)
= (l + â( ' M ' ) cot z + cot 3 z.
We may now write
r i (2) = 2-j Bn (e
(fc)
(fc)
—e
A
(fc)
), 11 & is odd,
2niz
\f/i (z) = B0 + 2-j Bn (e
-2niz
+e
), if k is even.
n=l
These expansions are valid in the strip defined by the inequalities
— 3 ( A T ) <$(Z) < 3 ( 7 T T ) . In either case, we have,
Blk) = (l/ir)
(6)
r'\[k)(Z)e-inUdz,
n* 0.
The value of B^ can be obtained from the relation
B(S)=-2±Bink)+
(7)
lim*^).
n=l
2—>0
To calculate the integral in (6), consider the contour integral
fö4'ii\z)e~2nif!dz where C is the boundary of the parallelogram with
vertices a t z~ ±T/2 and z = ±w/2 — (2S+\)TTT/2,
where s is an
arbitrary positive integer. This contour contains precisely 5 poles of
the function ^(z), these being located a t the points with affixes
z = —rwT> r = 1, 2, 3, • • • , s. On account of the properties of the functions involved we may pass to the limit as s tends to infinity, thus
obtaining
Bn
= — 2Î22, Rs
where R8 is the residue of \p[k)(z)e~2niz
z=
a t the poles of order kt
—SITT.
It is found in this manner that
(d)
B?-0;B?-^£.,
n*0.
1
-
<7 2 n
302
(e)
M. A. BASOCO
50
= 8 y.
rï
; -S» =
1 - q2"
(1 - g2")2
1 - g2"
>
«^0.
1 - g!
#/
8g2w
[April
{ « V w - (2?*2 - 6n - 6)q2n + (n2 - 6n + 6)},
On substituting the values of the coefficients given in (a), • • • , (f),
into (2) and (5) and transforming the resulting series into an arithmetical form, we find the following:
(I)
$o(s) = 4 Z qn{ Z sin 2fe}f
n = 1, 2, 3, 4, • • • ,
(n)
(II)
<j>x{z) = cot 2 + 4 X ? 2n { Z sin 2<fe},
(III)
*,(*) = - tan z + 4 X ) q2n{ Z ( - l ) d sin 2dz}9
(IV)
*,(*) = 4 E ( - 1)*?»{ £ sin 2fe},
(n)
(n)
(n)
(V)
^ ( s ) = 8 E gV(») + 8 £ / { £ (r - /) cos 2fe} f
(n)
(n)
(VI) tf J(s) = cot* * + 8 £ / > x ( » ) + 8 £ « ' " { £ (25 - d) cos 2</s},
(«)
02(2) = tan 2 + 8 2 ^ ?
(«)
£i(»)
(n)
- 8 £ ?'"{ Z ( - l) d (2« - <0 cos 2<fe},
(n)
(VIII) 03%) = 8 Z ( - 1)Y*(») + 8 Z ( - 1 ) V { Z (r - 0 cos 2/.},
(n)
(IX)
(n)
*&«) = W'ffî)4>*(z)
~ 4 £ / { £ (3r* - 6/r + 2/) sin 2/*},
(n)
8
</>i(z) = cot 2 + cot z + (${"/&{)4>i(z)
(X)
- 8 Z qn{ Z (<** - 6dÔ + 6Ô2) sin 2dz},
(n)
02(2) = - tan s - tan 3 0 +
(XI)
(#1'"/$l)fa{z)
- 8 Z / I Z (~ 1) V - MÔ + 6d2) sin 2&},
(n)
i943l
CLASS OF THETA QUOTIENTS
303
~ 4 Z ( " D Y { E (3r2 - 6*r + 2/) sin 2fe},
,YTn
(All;
(W)
(*i"7*i') = ~ l + 2 4 E A i W .
(n)
In these expansions, the notation is as usual: The inner sigma refers
to the divisors d, 8, t, r of the positive integer n, r being odd, and
cr(n) = ^ ( „ J - f ƒ (w)
where,
^i(w) = sum of the divisors of n whose conjugates are odd,
fi (n) = sum of the odd divisors of n, and
Ç\(n) = sum of all the divisors of n.
3. Paraphrases, The obvious identities
(8)
*«(*)•*«(*) = *!(*),
a = 0, 1, 2, 3,
in conjunction with the expansions (I) to (VIII) of the preceding section yield, through the method of paraphrase, 8 the following arithmetical theorems:
T H E O R E M (a). Let F(x) be an even, single-valued function which is
well defined f or all integral values of the argument including zero, but is
otherwise arbitrary ; let n be any positive integer. Then,
Z {FW - H - F(t' + t")} = E <T - ') {F(0 ~ F(0)},
where (k) and (I) refer to the following partitions of n in positive integers :
(*)
n = *V +
*"T",
(0
n = tr,
r, r',
T"
odd.
This is the arithmetical equivalent of (8) with a = 0. It is of interest
to note that a strictly elementary proof of this theorem has recently
been given by Uspensky. 3
THEOREM
£
{F{d'-d")
{i)
(/3). Let F(x) and the integer n be as in Theorem (a). Then,
-F(d'
+ d")}
= {fi(») - Un) }F(0) + £ (26 - d -
l)F(d)
(i)
- 2 £ {F(l)+F(2)+F(3) + . . - + F(<Z- 1)}
U)
8
E. T. Bell, Trans. Amer. Math. Soc. vol. 22 (1921) pp. 1-30; 198-219.
304
[April
M. A. BASOCO
where (i), (j) refer to the following partitions of n in positive integers :
(f)
n
= d'b' + d"b",
(j)
n = dô.
Ço(n) represents the number of positive integral divisors of n, while Çi(n)
represents their sum.
This theorem was first given by Liouville, 9 who stated it without
giving a proof; it is conjectured that he obtained this as well as other
similar results through the aid of the theory of elliptic functions. At
any rate, this theorem was later rediscovered by Bell (loc. cit.) as a
special case of a more general result derived from one of the addition
theorems of the theta functions and involving a certain set of doubly
periodic functions of the second kind. We have obtained this result
as the equivalent of the identity (8) with a = l.
T H E O R E M (7). Let F(x), n, and the partitions (i) and (j) be as in the
preceding theorem. Then,
£
( - !)*'+*"{p(d' - d") - F{d' + d"))
= {Un)
- Un)}F(0)
- E ( - 1)^(25 - d + l)F(d)
(j)
+ 2 Z {F(l)-F(2)+F(3)
+ ( - l)dF(d- 1)}.
(i)
This theorem is the arithmetical equivalent of identity (8) with
a = 2. It is, however, not essentially distinct from Theorem (/3); for
either theorem may be obtained from the other by replacing F(x) by
( — l)xF(x). Finally it should be noted that the case a = S merely yields
the result stated in Theorem (a).
As might be expected, those theorems analogous to the preceding
which may be deduced from identities involving the functions <j>\(z)
are considerably more complicated; it will perhaps suffice, in the
present instance, to state merely the following two results by way of
samples.
T H E O R E M (Ô). Let G(x) be an odd, single-valued function, which is
well defined for all integral values of the argument, but is otherwise arbitrary, let n be any positive integer. Then,
4 Z {G(h - h + h) + G(h + h - h) + G(h + h - h) - G(h + t2 + h)}
= 2 4 £ {Uni) L G{h) ) - E (3T* - 6tr + 2fi + 1)G(/),
(J)
9
Liouville, loc. cit.
(ft)
19431
CLASS OF THETA QUOTIENTS
305
where (i)f (j) and (k) refer to the following partitions of n in positive
integers :
(i) n = t\T\ + /2T2 + hr%
TI, T2, r 3 all odd,
(j)
n = 2ni + ^2»
(k)
n = /r,
ni = /2T2,
r 2 0<W,
r <?<&/.
This theorem is the arithmetical equivalent of the otherwise apparently trivial identity:
<t>o(z)-(j>o(z)'(l>o(z) = 4>o0).
T H E O R E M (e). £e£ F(x), n, and the partitions (i) and (j) be as in
Theorem (/3). Theny
6 E d'{d" - 2b") {F(d' + d") + F(d' - d")}
= 6(»fo(n) - fi(»))F(0) + 12 £ (8 - d){ £ 2 > M >
+ Z (d* - 6^2Ô + 6dô2 - d)F(â).
ij)
This theorem results on paraphrasing the identity:
d z
— #i0)
s
as
2 d
30i(2) — <£i0).
02
4. An application. If in Theorem (a) we place F(x) = x2, the following may be deduced
n-l
(9)
4 £ *i(*)*i(» - *) = * 8 (») - **i(*),
where ^ r (w) = s u m of the rth powers of the divisors of n whose conjugates are odd. In particular, if n is a prime p, we obtain
4 E Hk)Up - * ) = ( # - D(#2 - i).
A;=l
It also follows from (9) and the easily verified relationships:
nxn
^
1
™
X
n=l
2
n==l 1
2
" w (l + * »)*»
>
nzxn
"
"
_
>
#
ni/\\n)xn,
*
306
M. A. BASOCO
that,
j *
nxn
\2
nzxn
"
n2xn
*
2
l è î 1 - xH " è l 1 - * " "" e t (1 - x2-)2
- E-i
(i -
x 2 ") 2
Again, if in Theorem (/3) we place F(x) = x2, we deduce
n-X
(11)
12 Z fi(*)ri(» ~ *) = 5f,(f») - (6n -
l)fi(»),
which in turn implies the following development,
(12)
( °°
12{ Z
nxn
}2
°°
4 = £ (Sf.(«) - (6« -
Dfi(»))«".
Finally, in Theorem (5) place G(x) =x to deduce
24 E fiOOlhfo) = 2*8(») - (6» - l)^i(») + 3»r/(»),
W = 2fli + «2, »1, ^ 2 > 0.
This result may be interpreted so as to yield the following relation:
I zi i - W
(i4)
I zl i - *2»J
= 2T"
hZ
r i i - *«» z i i - *2w
*
+
3
èî
(2» -
6Z —
—
2n 2
z i (i - * )
l) 2 * 2 "- 1
(i - ***-i)» '
Results analogous to the preceding, but developed from an entirely
different point of view, have been obtained by Glaisher. 10 Here, such
results appear as very special consequences of the general theorems
stated in §3. It is clear, of course, that similar results may be obtained
through the full use of the expansions here given.
T H E UNIVERSITY OF NEBRASKA
10
J. W. L. Glaisher, Messenger of Mathematics nos. 166, 169 (1885).